Conjugate gradient algorithms are used for nonlinear optimization problems. Many different nonlinear problems can use conjugate gradient algorithms to find optimal solutions. One such nonlinear problem deals with tomography, which can be described as producing images of solid objects based on differences in the observed effects of waves of energy passing through the objects in sections, which is commonly used in radiology. Some examples of tomography include computed tomography (CT), single-photon emission tomography (SPET), positron emission tomography (PET), etc. In some more complicated iterative tomographic methods, tomographic reconstruction algorithms are used which may result in expansive problem sets using massive amounts of computing power to solve using conventional techniques.
Currently used analytical methods have difficulty with analyzing nonstandard geometries and nonuniform sampling data that is generated by tomographic data acquisition systems. Wide cone angles, linear and area array sources, and nonuniform sampling systems are not handled well by single pass reconstruction algorithms. Therefore, a general reconstruction method that is not dependent on any particular geometry, can incorporate a priori information, can give weight to more important data, and/or can incorporate regularization would be very beneficial to solving these issues. Such a methodology, however, has heretofore required vast amounts of computing power and/or resources in order to operate. Therefore, a methodology which does not expend vast amounts of computing power and/or resources to perform single pass reconstruction on these types of data sets would be very beneficial.